Higher-dimensional projector conjecture for smooth noncommutative Euclidean spaces

Let $P_k(A^\infty(\mathbb{R}^{2n}_\Theta))$ denote the class of projectors of level $k$ in the smooth noncommutative Euclidean algebra, where $k\geq0$, $n\geq1$, and $\Theta$ is the deformation parameter. The higher-dimensional projector conjecture.

$$P_k(A^\infty(\mathbb{R}^{2n}_\Theta))=\{0,1\}$$

for every $k\geq0$, every $n\geq1$, and every $\Theta$. This would classify all such projectors as trivial and is presented as a natural higher-dimensional conjecture; the source leaves it open.